A sailboat sails west from port for 5.3 km and experiences a shift in the wind direction. The wind blows the sailboat 2.5 km south. What is its displacement from its original position?
a^2+b^2=c^2
5.3^2+2.5^2=5.86^2
I just cant figure out the direction...
It must be southwest of its original position.
To find the displacement of the sailboat from its original position, you can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this scenario, the distance traveled west by the sailboat (5.3 km) and the distance traveled south (2.5 km) form the two sides of a right triangle. The displacement (c) represents the hypotenuse of the triangle.
Using the Pythagorean theorem, you can calculate the length of the displacement (c) as follows:
c^2 = 5.3^2 + 2.5^2
c^2 = 28.09 + 6.25
c^2 = 34.34
To find the value of c, you need to take the square root of both sides:
c = √34.34
c ≈ 5.86 km
So, the displacement of the sailboat from its original position is approximately 5.86 km.
Now, let's consider the direction. To determine the direction, you need to use trigonometry. In this case, you can use the tangent function.
The tangent of an angle in a right triangle is equal to the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle.
In this scenario, the angle you want to find is the angle between the displacement and the original position of the sailboat.
You can calculate the angle using the tangent function:
tan(θ) = opposite / adjacent
tan(θ) = 2.5 km / 5.3 km
θ = arctan(2.5 km / 5.3 km)
Using a calculator or mathematical software, you can find that θ is approximately 26.7 degrees.
Therefore, the displacement of the sailboat from its original position is approximately 5.86 km in a direction approximately 26.7 degrees south of west.